Exercise 2, § 3

With the notation of prop. 2 of I, p. 28 suppose that the relation $[\mathbf{a}.y] = 0$ for all $y \in F$ implies that $\mathbf{a} = 0$ in E. Under these conditions, if $g_i$ ($0 \leq i \leq n$) are $n + 1$ vector functions with values in E, defined on an interval I of $\mathbf{R}$ and such that for every vector function $\mathbf{f}$ with values in F and $n$ times differentiable on I, one has identically

then the functions $g_i$ are identically zero.